Biomedical Engineering Reference
In-Depth Information
the union of the supports of U and V is contained in an interval [a;b],
where 0 < a < b < 1 and 0 <
0
(a) <
0
(b) < 1.
(C3)
0
belongs to , a class of functions with bounded p-th derivative in
[a;b] for r 1 and the rst derivative of
0
is strictly positive and
continuous on [a;b].
(C4) The conditional density g(u;vjz) of (U;V ) given Z has bounded partial
derivatives with respect to (u;v). The bounds of these partial derivatives
do not depend on (u;v;z).
(C5) For some 2 (0; 1), a
T
var(ZjU)a a
T
E(ZZ
T
jU)a and
a
T
var(ZjV )a a
T
E(ZZ
T
jV )a a.s. for all a 2 R
d
.
It should be noted that in applications, implementation of the proposed
estimation method does not require these conditions to be satisfied. These
conditions are sucient but may not be necessary to prove the following
asymptotic theorem. Some conditions may be weakened but will make the
proof considerably more dicult. However, from a practical standpoint, these
conditions appear reasonable.
To study the asymptotic properties, we define a metric as follows: For any
1
;
2
2 , dene
k
1
2
k
2
= E[
1
(U)
2
(U)]
2
+ E[
1
(V )
2
(V )]
2
:
and for any
1
= (
1
;
1
) and
2
= (
2
;
2
) in the space of T = , dene
d(
1
;
2
) = k
1
2
k
T
=
k
1
2
k
2
+ k
1
2
k
2
1=2
:
Theorem 9.2: Let K
n
= O(n
), where satises the restriction 1=(1 + p) <
2 < 1=p. Suppose that T and (U;V ) are conditionally independent given Z
and that the distribution of (U;V;Z) does not involve (; ). Furthermore,
suppose that conditions (C1) through (C5) hold. Then,
(i) d (b
n
;
0
) = O
p
n
min(p;(1)=2)
. Thus if = 1=(1 + 2p), d(b
n
;
0
) =
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